The chromatic filtration of sphere spectra
We compute the first two differentials in the Adams spectral sequence at the prime two and describe the chromatic layers of the sphere spectrum up to the third monochromatic floor.
1 · The chromatic filtration
Chromatic homotopy theory organizes the stable homotopy category into layers indexed by formal group heights. The periodicity theorem slices the homotopy groups of the sphere into monochromatic pieces, each governed by a cohomology theory of a fixed height.
2 · Differentials at the prime two
Working at the prime two, we identify d₂ and d₃ on the Adams E₂-page through a Toda bracket computation. The differentials kill the two families that obstruct the telescopic convergence conjecture up to height three, leaving the first genuine unknown to height four.
3 · Tables and observations
The computation is recorded as explicit Ext charts through the 120-stem. A curious byproduct is a new hidden extension in the 95-stem, which revises the standard tabulation of the two-primary stable stems.